Finite Group Databasebeta

← C3×#10593294 #13166662

Subgroups — C3×#10593294 #13166662

Order 1728.

474 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64231C211yesmaximal, characteristic, complemented
2C3×#64227C211yesmaximal, characteristic, complemented
3C3×#64225C211yesmaximal, characteristic, complemented
4C3×#64221C211yesmaximal, characteristic, complemented
5C3×#63797C211yesmaximal, characteristic, complemented
6C3×#63796C211yesmaximal, characteristic, complemented
7C3×#63767C211yesmaximal, characteristic, complemented
8#10593294C311yesmaximal, characteristic, direct product factor
9C2×C3×GO+(4,𝔽2)—22no—
10C3×#3802C2211yescharacteristic
11C3×#3802C2211yescharacteristic
12C3×#3802C2211yescharacteristic
13C3×#3797C2211yescharacteristic
14C3×C4×#157C2211yescharacteristic
15C3×#3787C2211yescharacteristic
16C3×#3775—22no—
17C3×#3774—22no—
18C3×D32⋊C4—22no—
19C3×C8×C3⋊D3C2211yescharacteristic
20C32×D24—21no—
21C32×#204—21no—
22C3×#3715—21no—
23C3×#3713—21no—
24#64231C2×C311yescharacteristic, complemented
25#64227C2×C311yescharacteristic, complemented
26#64225C2×C311yescharacteristic, complemented
27#64221C2×C311yescharacteristic, complemented
28C3×D3×SD16—61no—
29C3×D3×D8—61no—
30#63797C2×C311yescharacteristic, complemented
31#63796C2×C311yescharacteristic, complemented
32#63767C2×C311yescharacteristic, complemented
33C2×C3×D32—21no—
34C2×C3×D32—21no—
35C2×C3×D32—21no—
36C2×C3×#157D412yes—
37C3×GO+(4,𝔽2)—44no—
38C3×C4×C3⋊D3C2311yescharacteristic
39C32×D12—21no—
40C32×D12—21no—
41C32×D12—21no—
42C32×Dic6—21no—
43C3×#627—21no—
44C3×#625—21no—
45C3×#625—21no—
46C3×#625—21no—
47C3×#624—21no—
48C3×#622D412yescomplemented
49C33×C8D411yescharacteristic, complemented
50C3×C3⋊(C3⋊C8)D411yescharacteristic, complemented
51C3×#496—91nomaximal
52C2×GO+(4,𝔽2)—22no—
53C3×D3×Q8—61no—
54C3×D3×D4—61no—
55C3×D3×D4—61no—
56C3×D3×D4—61no—
57#3802C22×C311yescharacteristic
58#3802C22×C311yescharacteristic
59#3802C22×C311yescharacteristic
60#3797C22×C311yescharacteristic
61C4×#157C22×C311yescharacteristic
62#3787C22×C311yescharacteristic
63#3775—22no—
64#3774—22no—
65D32⋊C4—22no—
66C32×SD16—61no—
67C32×D8—61no—
68C8×C3⋊D3C22×C311yescharacteristic
69C3×#215—61no—
70C3×#214—61no—
71C3×#213—61no—
72C3×#213—61no—
73C3×D24—21no—
74C3×D24—21no—
75C3×D24—41no—
76C3×#204—21no—
77C3×#204—21no—
78C3×#204—41no—
79C3×D3×C8—61no—
80C3×D3×C8—61no—
81#3715—21no—
82#3713—21no—
83C2×C3×C3⋊D3C2×D411yescharacteristic
84C2×C32×D3—41no—
85C2×C32×D3—41no—
86C2×C32×D3—41no—
87C3×D32—42no—
88C3×D32—42no—
89C3×D32—42no—
90C3×#157—22no—
91C33×C4C2×D411yescharacteristic
92C3×C3⋊Dic3C2×D411yescharacteristic
93C32×Dic3—41no—
94C2×C3×SD16—91no—
95C2×C3×SD16—91no—
96C2×C3×D8—91no—
97C2×C3×D8—91no—
98C3×#127—91no—
99C3×#124—91no—
100D3×SD16—61no—
101D3×D8—61no—
102C3×#97—91no—
103C22×C3×D3—121no—
104C22×C3×D3—121no—
105C22×C3×D3—121no—
106C2×D32—21no—
107C2×D32—21no—
108C2×D32—21no—
109C2×#157C3×D412yes—
110GO+(4,𝔽2)—44no—
111C32×Q8—61no—
112C32×D4—61no—
113C32×D4—61no—
114C32×D4—61no—
115C4×C3⋊D3C23×C311yescharacteristic
116C3×#67—121no—
117C3×#67—121no—
118C3×#67—121no—
119C3×D12—21no—
120C3×D12—21no—
121C3×D12—21no—
122C3×D12—21no—
123C3×D12—21no—
124C3×D12—21no—
125C3×D12—41no—
126C3×D12—41no—
127C3×D12—41no—
128C3×C4×D3—61no—
129C3×C4×D3—61no—
130C3×C4×D3—121no—
131C3×Dic6—21no—
132C3×Dic6—21no—
133C3×Dic6—41no—
134C3×Dic6—121no—
135#627—21no—
136#625—21no—
137#625—21no—
138#625—21no—
139#624—21no—
140#622C3×D412yescomplemented
141C32×C8—21no—
142C32×C8—21no—
143C32×C8—41no—
144C32×C8—41no—
145C32×C8C3×D411yescharacteristic, complemented
146C3⋊(C3⋊C8)C3×D411yescharacteristic, complemented
147C3×C3⋊C8—61no—
148C3×C3⋊C8—61no—
149#496—91noSylow
150C2×C33#12711yescharacteristic
151C3×C3⋊D3AGL(1,ℤ/8)12yes—
152C32×D3—81no—
153C32×D3—81no—
154C32×D3—81no—
155C2×C3×Q8—91no—
156C2×C3×D4—91no—
157C2×C3×D4—91no—
158C2×C3×D4—91no—
159C2×C3×D4—182no—
160D3×Q8—61no—
161D3×D4—61no—
162D3×D4—61no—
163D3×D4—61no—
164C3×SD16—61no—
165C3×SD16—61no—
166C3×SD16—121no—
167C3×SD16—181no—
168C3×SD16—182no—
169C3×D8—61no—
170C3×D8—61no—
171C3×D8—121no—
172C3×D8—181no—
173C3×D8—182no—
174C2×C3×C8—91no—
175C2×C3×C8—91no—
176C3×C22⋊C4—182no—
177C3×C42—91no—
178#215—61no—
179#214—61no—
180#213—61no—
181#213—61no—
182D24—21no—
183#204—21no—
184D3×C8—61no—
185D3×C8—61no—
186C22×C32—121no—
187C22×C32—121no—
188C22×C32—121no—
189C2×C3⋊D3C2×C3×D411yescharacteristic
190C2×C3×D3—41no—
191C2×C3×D3—41no—
192C2×C3×D3—41no—
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—81no—
199C2×C3×D3—81no—
200C2×C3×D3—81no—
201C2×C3×D3—242no—
202C2×C3×D3—242no—
203C2×C3×D3—242no—
204D32—42no—
205D32—42no—
206D32—42no—
207#157—22no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—41no—
211C32×C4—41no—
212C32×C4—121no—
213C32×C4C2×C3×D411yescharacteristic
214C3⋊Dic3C2×C3×D411yescharacteristic
215C3×Dic3—41no—
216C3×Dic3—41no—
217C3×Dic3—61no—
218C3×Dic3—61no—
219C3×Dic3—81no—
220C2×SD16—91no—
221C2×SD16—91no—
222C2×D8—91no—
223C2×D8—91no—
224#127—91no—
225#124—91no—
226#97—91no—
227C33#49611yescharacteristic, complemented, Sylow
228C23×C3—181no—
229C23×C3—181no—
230C23×C3—181no—
231C22×D3—121no—
232C22×D3—121no—
233C22×D3—121no—
234C3×Q8—61no—
235C3×Q8—61no—
236C3×Q8—121no—
237C3×Q8—181no—
238C3×D4—61no—
239C3×D4—61no—
240C3×D4—61no—
241C3×D4—61no—
242C3×D4—61no—
243C3×D4—61no—
244C3×D4—121no—
245C3×D4—121no—
246C3×D4—121no—
247C3×D4—181no—
248C3×D4—181no—
249C3×D4—181no—
250C3×D4—364no—
251C2×C3×C4—91no—
252C2×C3×C4—92no—
253C2×C3×C4—181no—
254#67—121no—
255#67—121no—
256#67—121no—
257D12—21no—
258D12—21no—
259D12—21no—
260C4×D3—61no—
261C4×D3—61no—
262C4×D3—121no—
263Dic6—21no—
264Dic6—121no—
265C3×C8—21no—
266C3×C8—21no—
267C3×C8—41no—
268C3×C8—41no—
269C3×C8—92no—
270C3×C8—91no—
271C3×C8GO+(4,𝔽2)11yescharacteristic, complemented
272C3⋊C8—61no—
273C3⋊C8—61no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—241no—
279C2×C32—241no—
280C2×C32—241no—
281C2×C32C3×#12711yescharacteristic
282C3⋊D3C3×AGL(1,ℤ/8)12yes—
283C3×D3—62no—
284C3×D3—62no—
285C3×D3—81no—
286C3×D3—81no—
287C3×D3—81no—
288C3×D3—81no—
289C3×D3—81no—
290C3×D3—81no—
291C3×D3—161no—
292C3×D3—161no—
293C3×D3—161no—
294C2×Q8—91no—
295C2×D4—91no—
296C2×D4—91no—
297C2×D4—91no—
298C2×D4—182no—
299SD16—61no—
300SD16—181no—
301SD16—182no—
302D8—61no—
303D8—181no—
304D8—182no—
305C2×C8—91no—
306C2×C8—91no—
307C22⋊C4—182no—
308C42—91no—
309C22×C3—91no—
310C22×C3—121no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3—121no—
315C22×C3—121no—
316C22×C3—241no—
317C22×C3—241no—
318C22×C3—241no—
319C22×C3—362no—
320C22×C3—362no—
321C22×C3—362no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—61no—
326C2×D3—61no—
327C2×D3—242no—
328C2×D3—242no—
329C2×D3—242no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—91no—
335C3×C4—121no—
336C3×C4—121no—
337C3×C4—182no—
338C3×C4—241no—
339C3×C4C2×GO+(4,𝔽2)11yescharacteristic
340Dic3—41no—
341Dic3—61no—
342Dic3—61no—
343C32—21no—
344C32—21no—
345C32—41no—
346C32—41no—
347C32C3×#49611yescharacteristic, complemented
348C23—181no—
349C23—181no—
350C23—181no—
351Q8—61no—
352Q8—181no—
353D4—61no—
354D4—61no—
355D4—61no—
356D4—181no—
357D4—181no—
358D4—181no—
359D4—364no—
360C2×C4—91no—
361C2×C4—92no—
362C2×C4—181no—
363C8—92no—
364C8—91no—
365C8C3×GO+(4,𝔽2)11yescharacteristic, complemented
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—92no—
371C2×C3—241no—
372C2×C3—241no—
373C2×C3—241no—
374C2×C3—241no—
375C2×C3—241no—
376C2×C3—241no—
377C2×C3—481no—
378C2×C3—481no—
379C2×C3—481no—
380C2×C3#6423111yescharacteristic
381D3—62no—
382D3—62no—
383D3—81no—
384D3—81no—
385D3—81no—
386C22—91no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—362no—
391C22—362no—
392C22—362no—
393C4—91no—
394C4—121no—
395C4—182no—
396C4C2×C3×GO+(4,𝔽2)11yescharacteristic
397C3—21no—
398C3—21no—
399C3—41no—
400C3—41no—
401C3#1059329411yescharacteristic, direct product factor
402C2—92no—
403C2—241no—
404C2—241no—
405C2—241no—
406C2C3×#6423111yescharacteristic
subgroupG/H
centerC2×C3#64231
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#127
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#496
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#64231C2semidirect
2. C3×#64227C2semidirect
3. C3×#64225C2semidirect
4. C3×#64221C2semidirect
5. C3×#63797C2semidirect
6. C3×#63796C2semidirect
7. C3×#63767C2semidirect
8. #10593294C3direct (N × H)
24. #64231C2×C3semidirect
25. #64227C2×C3semidirect
26. #64225C2×C3semidirect
27. #64221C2×C3semidirect
30. #63797C2×C3semidirect
31. #63796C2×C3semidirect
32. #63767C2×C3semidirect
48. C3×#622D4semidirect
49. C33×C8D4semidirect
50. C3×C3⋊(C3⋊C8)D4semidirect
140. #622C3×D4semidirect
145. C32×C8C3×D4semidirect
146. C3⋊(C3⋊C8)C3×D4semidirect
227. C33#496semidirect
271. C3×C8GO+(4,𝔽2)semidirect
347. C32C3×#496semidirect
365. C8C3×GO+(4,𝔽2)semidirect
401. C3#10593294direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#642312
C3×#642272
C3×#642252
C3×#642212
C3×#637972
C3×#637962
C3×#637672
#105932943
C3×#4969